Top-level dynamics and the regulated gene response of feed-forward loop transcriptional motifs

Feed-forward loops are hierarchical three-node transcriptional subnetworks, wherein a top-level protein regulates the activity of a target gene via two paths: a direct-regulatory path, and an indirect route, whereby the top-level proteins act implicitly through an intermediate transcription factor. Using a transcriptional network of the model bacterium Escherichia coli, we confirmed that nearly all types of feed-forward loop were significantly overrepresented in the bacterial network. We then used mathematical modeling to study their dynamics by manipulating the rise times of the top-level protein concentration, termed the induction time, through alteration of the protein destruction rates. Rise times of the regulated proteins exhibited two qualitatively different regimes, depending on whether top-level inductions were “fast” or “slow.” In the fast regime, rise times were nearly independent of rapid top-level inductions, indicative of biological robustness, and occurred when RNA production rate-limits the protein yield. Alternatively, the protein rise times were dependent upon slower top-level inductions, greater than approximately one bacterial cell cycle. An equation is given for this crossover, which depends upon three parameters of the direct-regulatory path: transcriptional cooperation at the DNA-binding site, a protein-DNA dissociation constant, and the relative magnitude of the top-level protien concentration.

Figure 1

(a) All eight feed-forward loop transcriptional motifs. The composite sign of the indirect path, (1,2) and then (2,3), is identical to the direct path (1,3) for the “coherent” loops, but is opposite for the “incoherent” loops [10]. The regulatory action of the single (2,3) link differentiates the coherent from the incoherent loops of each type. The search algorithm included counts of (b) embedded feed-forward loops or excluded them to count only the (c) canonical loops.

Figure 2

(a) Concentrations for the top-level regulator, ${\tilde{S}}_1\left(\tilde{t}\right)$, of each feed-forward loop, under conditions of varying induction time values, ${\tilde{\tau }}_1$. Steplike activation of the node 1 regular (dotted line), the point at which all other expression responses are measured, confers a graded response in the node 1 protein production (solid lines). Without manipulation, induction time is fixed, ${\tilde{\tau }}_1=ln2$ (thick solid line). Here, $n=2$ and ${\tilde{S}}_{\max }=10$. (b) Measurements of rise times for protein concentrations in the feed-forward loop from a steplike activator of node 1 (wavy line).

Figure 3

Rise times for the four coherent (red) and incoherent (blue) loops are plotted against induction times for the top-level regulator node 1 (see Fig. 1), and illustrated on a logarithmic scale. Dotted (incoherent FFLs) or solid (coherent FFLs) lines show best-fit solutions to a linear model. The solid black line denotes ${\tilde{\tau }}_3={\tilde{\tau }}_1$.

Figure 4

(a) Pulsing dynamics of the I-3 feed-forward loop. (b) Comparison in normalized response between the full I-1 loop and the direct linkage of the I-1 loop. (c) Ratio of rise time in a feed-forward loop, ${\tilde{\tau }}_3\left(\text{FFL}\right)$, to that of the single, direct linkage, ${\tilde{\tau }}_3\left(\text{single}\right)$, for coherent (red) and incoherent (blue) feed-forward loops. Values calculated using $n=2$, ${\tilde{S}}_{\max }=10$, and all $\tilde{K}=2.81$.

Figure 5

Diagram for regimes of the robustness threshold, ${\tilde{\tau }}^R$ [Eq. (13)]. Regions $\mathrm{I}$ and $\mathrm{II}$ illustrate parameters associated with a responsiveness to perturbation, and those of region $\mathrm{III}$ lead to approximately unresponsive dynamics. The shaded area, region $\mathrm{II}$, shows parameter values such that ${\tilde{\tau }}^R$ resides within one cell cycle (fixed $n=2$).

Figure 6

Illustration of the three-regime piecewise approximations for ${\tilde{f}}_j\left(\tilde{t}\right)$ and ${\tilde{S}}_i\left(\tilde{t}\right)$. Both axes are logarithmically scaled, wherein straight lines correspond to power laws.

Figure 7

Plot of the exponent $\xi$ against ${\tilde{\tau }}_j$ for various values for the induction times, ${\tilde{\tau }}_i$. Horizontal line indicates the position of the right-hand side of Eq. (B14).

Figure 8

Plot of rise time, ${\tilde{\tau }}_j$, against the induction time, ${\tilde{\tau }}_i$, obtained from numerical integration of Eq. (B2) (circles) or from the analytic formula of Eq. (B21) (line).